The solution(s) of $15 {(2)}^{x + 1} + 15 {(2)}^{2 - x} = 135$ is/are

2

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1

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Solution

The correct options are
A $2$
D $- 1$
This equation can be rewritten in the form
$30. 2^{x} + \frac{60}{2^{x}} = 135$

Let $t = 2^{x}$ , then

${30 t}^{2} - 135 t + 60 = 0 {6 t}^{2} - 27 t + 12 = 0$

${\Rightarrow 6 t}^{2} - 24 t - 3 t + 12 = 0 \Rightarrow (t - 4) (6 t - 3) = 0$
then $t_{1} = 4$ and $t_{2} = \frac{1}{2}$

Thus given equation is equivalent to

$2^{x} = 4$ and $2^{x} = \frac{1}{2}$

$⟹ x_{1} = 2$ and $x_{2} = - 1$

Hence roots of the original equation are $2$ and $- 1$

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